2.1. Modeling the Uncertainty Using a Stochastic Tree
Many of the parameters used to model the generation system can be considered as known input data such as the nominal technical characteristics of the generating units (maximum power, input-output curves, etc.) or the initial value of the hydraulic reserves. However, there are many other parameters which are subject to uncertainty. Among them, the ones linked to the meteorology are crucial. On the one hand, the amount of rain or snow affects the level of hydroelectric energy stored in the reservoirs. In fact, the possible scenarios of natural hydraulic inflows constitute one of the most significant concerns when planning the medium term operation of the system. Other examples are the wind speed that determines the power that can be produced in wind farms, or solar radiation conditions for electrical production in photovoltaic or thermo-solar installations. On the other hand, meteorological factors also affect the electricity demand, which also depends on the economic activity and working patterns. Finally, the fuel prices of conventional thermal generators play a key role when determining the optimal dispatch of the generators, and the fluctuations in the coal and natural gas markets are also important sources of uncertainty.
A common approach for taking into account this uncertainty within an optimization model is to adopt a discrete representation of the probability distribution of all the random parameters in the form of a multistage scenario tree that considers the non-anticipative criterion of the decisions, (see [
20]).
Figure 1 shows an example of a multistage scenario tree that consists of
nodes.
Each one of the nodes is denoted by where is the time stage and is the position of the node at that time period. Given that every node as a single predecessor, it is common to use the term father node that is indicated by . For instance . In addition, the descendants of each node are denoted by . For example, .
Notice that the first node, i.e., the root node denoted as
, is unique and it represents a
here and now decision node as the value of the decision variables corresponding to it will be the same for all the scenarios. A typical example of this kind of decisions is the problem of determining the level of the reservoirs at the end of the first time-period knowing that the future natural inflows are uncertain. The posterior decision variables are
recourse functions as they can be adapted as far as the uncertainty is being unveiled. All the branches can be characterized by a certain probability of occurrence, which leads to a total probability of occurrence of each scenario
that satisfies that
. Notice that each scenario
is made of a whole set of nodes starting at the root node, and finishing in a terminal node, i.e., a node without any descendant. The set of nodes that belong to a given scenario
are denoted as
. For instance, the nodes that correspond to scenario
have been highlighted in
Figure 1. In addition, every node can also be characterized by a probability of occurrence denoted by
.
2.2. Hydroelectric Generation
The power generated by a hydro unit depends on the water flow impacting the blades of the turbine and the net head, i.e., the difference between the elevation of the stored water and the elevation of the turbine drain, minus energy losses within the pipeline. In addition, the output power also depends on the efficiency of the turbine, the drive system and the generator. In order to model these relationships, it is common to use a family of input–output curves relating the water flow and the output power for each possible net-head value [
21]. These input–output curves are very common in short-term hydro scheduling models [
22], where the hourly generation of the hydro plants needs to be carefully modeled. However, in medium-term models it is usual to aggregate the production of many hours into a representative time period (for instance, all the peak hours of the working days of a given week). In this case, instead of the instantaneous relationship between power and flow rate, it is more relevant to model the ratio between the energy produced during such aggregated time-periods and the total volume of water released through the hydro turbine. This ratio is named as the
energy coefficient, and for a given hydro plant
, it is denoted by
. Instead of assuming a static representation (as in [
17]), this paper considers its dependency on the stored volume of water given that the net head changes with respect to the volume of water stored accordingly to the shape of the pond.
2.3. Mathematical Formulation of the Centralized Model
The objective of a centralized planner is to find the operation of the system that minimizes the expected cost while satisfying the demand balance equation and all the technical constraints of the system. The objective function can be formulated as
Notice that in (1), the costs of the thermal generators at each node are multiplied by the probability and the duration of the corresponding node. In addition, the per-hour cost functions of each generator depend on the power produced by them , and they can be different for each node in order to allow different fuel-price scenarios.
The optimization problem is subject to the following set of constraints:
Equation (2) establishes that the sum of the production of all the thermal generators () plus the sum of all hydroelectric units () plus the generation of renewable energy sources () must satisfy the demand at every node. Notice that for each constraint, its corresponding Lagrange multiplier is shown after the symbol that indicates complementarity. The Lagrange multiplier of (2) is represented by and it measures what would be the impact on the objective function if the demand in this particular node increases by one unit.
Equation (3) establishes the water balance equation: the volume of stored water at the reservoir of hydro unit at the end of the time stage that corresponds to the node is equal to the volume of water at the end of the previous period defined by the father node, , plus the amount of water that corresponds to the natural inflows , minus the amount of water due to the water flow released to generate hydro power, , minus the possible spillages, . In the particular case of the root-node, the volume of the father node is the initial volume, , which is input data. In addition, for the particular case of the terminal nodes, , the volume stored will not be a variable of the problem but the predefined target value at the end of the planning horizon, i.e., .
The relationship between the water flow discharged from the hydro turbine, , and the output power generated, , is expressed in (4) where is the energy coefficient that depends on the volume of water stored at the reservoir. The water flow limits of hydro units are established in (5) to model the physical limits of the intake of the turbine or any other water right such as minimum ecological flows. Notice that the generation limits of the hydro units are a consequence of the joint consideration of Constraints (4) and (5). However, in case the electric generator had a more restrictive power limit, it would be possible to add a new constraint imposing such limit. The maximum and minimum limits for thermal generators are taken into account in (6). Notice that, for the sake of simplicity, the existence of minimum stable loads for thermal generators has not been considered as this would require the usage of binary variables. As the presented model is intended to illustrate the impact of risk aversion levels on the operation of the hydro reservoirs in the medium term, all the issues related to a detailed modelling of thermal generators and their intertemporal constraints (such as ramps), have been neglected. The limits of the reservoirs are included in (7) which are formulated for all the nodes of the tree except for the terminal nodes, given that as it was mentioned before, at the last stage the volumes are fixed to the target level which is supposed to be feasible without any loss of generality. Finally, the non-negativity constraint of spillages is formulated in (8).